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Physical Review A

Transient four-wave and six-wave mixing with the excitation of subharmonic Raman resonances

Nitin C. Kothari, G. S. Agarwal

Published Aug 1, 1990• 1 citation
Publisher

Abstract

Molecular polarizability theory with the inclusion of the second-hyperpolarizability coefficient is used to study the transient generation of four-wave- and six-wave-mixing signals when a system interacts with laser and Stokes pulses of frequencies ${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathit{S}}$ lying in the region of the subharmonic Raman resonance ${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}$-${\mathrm{\ensuremath{\omega}}}_{\mathit{S}}$\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{R}}$/2. The dynamical equations for various fields and amplitudes of the Raman and the subharmonic mode are derived and integrated numerically to obtain the detailed characteristics, such as pulse width, pulse center, ringing, etc., of the generated signals. The effect of the chirp in the incident pump pulse is also studied.

DOI
10.1103/physreva.42.1775

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Faraday Discussions•Jan 1, 1999•22 citations
Molecular polarizability theory with the inclusion of the second-hyperpolarizability coefficient is used to study the transient generation of four-wave- and six-wave-mixing signals when a system interacts with laser and Stokes pulses of frequencies \ensuremathωl{\mathrm{\ensuremath{\omega}}}_{\mathit{l}}\ensuremathωl​ and \ensuremathωS{\mathrm{\ensuremath{\omega}}}_{\mathit{S}}\ensuremathωS​ lying in the region of the subharmonic Raman resonance \ensuremathωl{\mathrm{\ensuremath{\omega}}}_{\mathit{l}}\ensuremathωl​-\ensuremathωS{\mathrm{\ensuremath{\omega}}}_{\mathit{S}}\ensuremathωS​\ensuremath{\approxeq}\ensuremathωR{\mathrm{\ensuremath{\omega}}}_{\mathit{R}}\ensuremathωR​/2. The dynamical equations for various fields and amplitudes of the Raman and the subharmonic mode are derived and integrated numerically to obtain the detailed characteristics, such as pulse width, pulse center, ringing, etc., of the generated signals. The effect of the chirp in the incident pump pulse is also studied.
A general theory of transient stimulated Raman scattering and the phase-matching diagrams are developed based on an interaction Hamiltonian which includes all the possible competing interaction channels. In a Raman system pumped by multiple crossing beams with multiple Stokes and anti-Stokes components, these channels are defined by the interactions among fields at different frequencies and/or k vectors of the pump pairs and Stokes and anti-Stokes fields. An ensemble-averaged equation of motion for the active phonon mode is derived within the Markovian approximation in which the phonon-phonon coupling (pressure effect) is reduced to an energy relaxation and a phase fluctuation. A complete set of transient coupled equations including phase modulation, Stark shift, medium (anharmonic) saturation, and all the possible four-wave-mixing channels is presented. In the transient regime, it is found that the effects of the Stark shift on the Raman efficiency are mainly governed by the input pump energy in contrast to the steady-state theory where the pump intensity plays the main role. Both analytic and numerical methods are developed in describing the nonlinear effects due to the energy feedback mechanism in the strong pump-depletion regime. Twelve possible four-wave-mixing channels are analyzed for nonlinear phase diagrams characterized by the angles among pump pairs, and Stokes and anti-Stokes fields. Finally, numerical examples are shown for the effects of the coherence lengths of high-order Stokes fields and pump pairs on the generation of low-order Stokes fields.
We will demonstrate that four-wave mixing with linearly chirped (phase-modulated) pulses is a unique tool for obtaining information on the dynamics and level structure of a system. Especially, it will be shown that the transient-grating-scattering type of experiment with chirped pulses provides an immediate answer to the question of whether the dynamics of a system occurs on a fast and/or slow time scale. In addition, we present compelling evidence that chirped four-wave mixing in a molecular system is a viable method for measuring excited-state vibrational frequencies. Double-sided Feynman diagrams are used for a third-order perturbative calculation of two-level four-wave-mixing effects and chirped coherent Raman scattering. The diagrams provide a visual representation of the quantum-mechanical pathways that the system can take as a result of the different field interactions. The number of quantum-mechanical pathways that contribute to the signal is shown to depend on the chirp rate compared to the time scale(s) of the system dynamics. A stochastic model is used to describe the optical dynamics of the system. The resulting expressions for the third-order nonlinear polarization are so complex that numerical calculations are necessary to simulate the time dependence of the optical response. It will also be shown that our theoretical results in the appropriate limits converge to those obtained for impulsive or continuous-wave excitation.
The delayed or time-domain four-wave-mixing experiment is treated in the regime of intense near-resonant pulses. The interaction with the radiation during both pump and probe pulses is considered to all powers of the electric field amplitude. Analytical results are obtained for an effective four-level system. These include the dependence of the coherence amplitudes on the ratio of the pump-field intensities when there is a large vibrational discrepancy between ground and excited electronic states and a general solution for the unitary time development during the probe pulse. For the first time, delayed coherent anti-Stokes Raman scattering is detected from highly dilute (10-ppm) guest molecules. Illustrative examples are presented for the system of pentacene in benzoic acid at low temperature. Vibronic-free induction decay and the effect of field inhomogeneity across the beam profile are found to be essential for understanding the observed intensity and spectral distribution of the signal beam in the region of optimum pulse intensity.
The existence of odd Raman subharmonics in the nonlinear mixing of fields of frequencies ${\mathrm{\ensuremath{\omega}}}_{\mathit{l}}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathit{s}}$ is examined in a nonperturbative way. It is shown that the coupling of Stokes and anti-Stokes processes at high pump powers produces odd subharmonics in nonlinearly generated signals. It is also shown that this coupling can lead to gain at the odd subharmonics. An effective Hamiltonian is used to discuss the coupling of Stokes--anti-Stokes processes. Numerical results showing the odd subharmonics are presented. The origin of odd subharmonics in terms of transitions among dressed states is also discussed.
Three-pulse four-wave mixing (FWM) is used here to study and control laser excitation processes. For general laser excitation processes, after a molecule interacts resonantly with a laser pulse, the molecule has a probability of being in the ground or in the excited state. Control over this process depends on the phase and amplitude of the electric fields that interact with the molecular system. Here we show how three-pulse FWM can be used to control the excitation of iodine molecules. Depending on the time delay between the first two pulses, the observed signal reflects the dynamics of the ground or excited state. A theoretical formalism based on the density matrix formulation is presented and solved for a four-level system. Experiments are found to be in excellent agreement with the theory. The influence of linear chirp on three-pulse FWM experiments is explored. Spectrally dispersed three-pulse FWM is found to be extremely useful for studying the effect of chirp on laser excitation of molecular systems. Experimental demonstrations of these effects are included.